Geometry Chapter 10-Circles

In Theorem 10.12, the phrase_________ is equivalent to saying that the " circle is circumscribed about the quadrilateral."

"a quadrilateral... inscribed in a circle"

Products of the lengths of collinear segments from a point of intersection to the circle are equal.

(DC)(CB)=(CA)(CE) (DE)(DA)=(DC)(DB) (CD)(CA)=(CB)2 point to circle x point to circle=point to circle x point to circle

The standard equation of a circle with radius r and center (h,k) is

(x-h)2+(y-k)2=r2

circle

1. derived from the Latin word circus, which means "ring" or "racecourse" 2. is the set of all points in a plane that are equidistant from a given point, called the center of the circle.

The procedure for finding the equation of a circle that passes through three points is summarized as follows:

1.Consider the triangle formed by the three points. Draw the perpendicular bisectors of two of the sides. 2.The center of the circle is the point of intersection of the two perpendicular bisectors. 3.The radius of the circle is the distance between the center and any of the three given points. 4.Use the center and radius to write the standard equation of the circle.

Theorem 10.12

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

Theorem 10.11

An angle that is inscribed in a circle is a right angle if and only if its corresponding arc is a semicircle.

diameter

a chord that passes through the center

radius

a segment that has the center as one endpoint and a point on the circle as the other endpoint

chord

a segment whose endpoints are on the circle

Central Angle of a Circle

an angle whose vertex is the center of the circle

A circle is ______ about a polygon if each vertex of the polygon lies on the circle.

circumscribed

A common tangent that does not intersect the segment that joins the centers of the circles is a_______

common external tangent

A common tangent that intersects the segment that joins the centers is a _________

common internal tangent

A line that is tangent to two circles is a ______ of the circles

common tangent

Circles that have the same center are called_____

concentric

In the same circle,or in congruent circles, two arcs are ______ if they have the same measure

congruent

Two circles are ______ if they have congruent radii or congruent diameters.

congruent

The common length of all diameters of a circle is the

diameter of the circle

Two arcs of the same circle are adjacent if they intersect at ________

exactly one point

The points outside the circle form the circle's

exterior

A circle is _____ in a polygon if each side of the polygon is tangent to the circle

inscribed

An angle, angle ABC is an _______ of a circle if line segment AB and line segment BC are chords of the circle

inscribed angle

The arc that lies in the interior of an inscribed angle is the ______ of the angle.

intercepted arc

The points inside the circle form the circle's

interior

The ________ associated with a central angle angle APB consists of the points A and B and all points of circle P that lie in the exterior of angle APB.

major arc

The __________ is defined to be the difference between 360 degrees and the measure of its associated minor arc.

measure of a major arc

The _________________ is defined to be the measure of its central angle.

measure of a minor arc

When the vertex of the angle is in the exterior of the circle measure of angle=1/2 difference of the intercepted arcs

measure of angle 1= 1/2(measure of minor arc CD-measure of minor AB) measure of angle 1= 1/2(measure of minor arc AC- measure of minor arc AB) measure of angle 1= 1/2(measure of major arc ACB-measure of minor arc AB)

When the vertex of the angle is on the circle, measure of angle=1/2 measure of the the arc intercepted by the angle.

measure of angle 1= 1/2(minor arc AB)

When the vertex of the angle is in the interior of the circle, measure of angle= 1/2 sum of the intercepted arcs

measure of angle 1= measure of angle 2= 1/2(measure of minor arc AB+measure of minor arc CD)

If measure of angle APB less than 180 degrees, then the points A and B, together with the points of circle P that lie in the interior of angle APB, form a ________ of the circle

minor arc

The point at which it intersects the circle is the

point of tangency

By the the definition of a circle, all ____ of a circle are congruent

radii

If a line intersects a circle at two points, then the line is a _____ of the circle.

secant

If the endpoints of an arc are the endpoints of a diameter, then the arc is a ________ and its measure is 180 degrees

semicircle

If a line intersects a circle at exactly one point, then the line is a _______ of the circle.

tangent

Two circles can have exactly one point of intersection. Such circles are ______ to each other.

tangent

center of a circle

the point inside the circle that is equidistant from all the points on the circle

Their common length is called

the radius of the circle

Major arcs (and semicircles) are denoted by ______ letters

three

The diameter of a circle is _____ its radius

twice

A line can intersect a circle in ____ ways.

two

Minor arcs are denoted by _____ letters

two

distance formula for circle

√(x-h)2+(y-k)2

distance formula

√(x2-x1)2 + (y2-y1)2

Theorem 10.6

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

Theorem 10.1

If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

Theorem 10.13

If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is half the measure of its intercepted arc.

Theorem 10.15

If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of the intercepted arcs.

Theorem 10.9

If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.

Theorem 10.7

If chord line segment AB is a perpendicular bisector of another chord then line segment AB is a diameter.

Theorem 10.14

If two chords intersect in the interior of a circle, then the measure of each angle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Theorem 10.10

If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

Theorem 10.3

If two segments from the same exterior point are tangent to a circle, then they are congruent.

Theorem 10.2

In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

Theorem 10.8

In the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

Theorem 10.5

In the same circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

Theorem 10.4

In the same circle, or in congruent circles, two arcs are congruent if and only if their central angles are congruent.

Postulate 21

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs